Daniel is packing his bags for his vacation. He has $5$ unique toy animals, but only $3$ fit in his bag. How many different groups of $3$ toy animals can he take?
Answer: Daniel has $3$ spaces for his toy animals, so let's fill them one by one. At first, Daniel has $5$ choices for what to put in the first space. For the second space, he only has $4$ toy animals left, so there are only $4$ choices of what to put in the second space. So far, it seems like there are $5 \cdot 4 = 20$ different unique choices Daniel could have made to fill the first two spaces in his bag. But that's not quite right. Why? Because if he picked toy animal number 3, then toy animal number 1, that's the same situation as picking number 1 and then number 3. They both end up in the same bag. So, if Daniel keeps filling the spaces in his bag, making $5\cdot4\cdot3 = \dfrac{5!}{(5-3)!} = 60$ decisions altogether, we've overcounted a bunch of groups. How much have we overcounted? Well, for every group of $3$, we've counted them as if the order we chose them in matters, when really it doesn't. So, the number of times we've overcounted each group is the number of ways to order $3$ things. There are $3! = 6$ ways of ordering $3$ things, so we've counted each group of $3$ toy animals $6$ times. So, we have to divide the number of ways we could have filled the bag in order by number of times we've overcounted our groups. $ \dfrac{5!}{(5 - 3)!} \cdot \dfrac{1}{3!}$ is the number of groups of toy animals Daniel can bring. Another way to write this is $ \binom{5}{3} $, or $5$ choose $3$, which is $10$.